We are given the profit function:

$P(x) = -0.1x^3 + 200x - 70$

We want to use differentials to approximate the change in profit for an increase in production (or sales) of one unit. To do this, we will follow these steps:

Step 1: Find the derivative of the profit function.

The derivative of $P(x)$ with respect to $x$ will represent the rate of change of profit with respect to $x$, i.e., how the profit changes when $x$ changes.

$P'(x) = \frac{d}{dx} \left( -0.1x^3 + 200x - 70 \right)$

Calculating the derivative:

$P'(x) = -0.3x^2 + 200$

Step 2: Use the derivative to approximate the change in profit.

The differential $dP$ is given by:

$dP = P'(x) \, dx$

where $dx$ is the change in $x$. In this case, we are interested in the change when $x$ increases by one unit, so $dx = 1$.

Now, plug in $x = 50$ into $P'(x)$ to find the rate of change at that point:

$P'(50) = -0.3(50)^2 + 200 = -0.3(2500) + 200 = -750 + 200 = -550$

Thus, the differential approximation for the change in profit is:

$dP = P'(50) \cdot 1 = -550 \cdot 1 = -550$

Step 3: Find the actual change in profit.

Now, calculate the actual change in profit when $x$ increases from $x = 50$ to $x = 51$. This requires evaluating the profit at $x = 51$ and $x = 50$.

$P(51) = -0.1(51)^3 + 200(51) - 70$

$P(51) = -0.1(132651) + 10200 - 70 = -13265.1 + 10200 - 70 = -3065.1$

$P(50) = -0.1(50)^3 + 200(50) - 70 = -0.1(125000) + 10000 - 70 = -12500 + 10000 - 70 = -2570$

The actual change in profit is:

$\Delta P = P(51) - P(50) = -3065.1 - (-2570) = -3065.1 + 2570 = -495.1$

Step 4: Compare the differential approximation with the actual change.

• The differential approximation for the change in profit is $-550$.
• The actual change in profit is $-495.1$.

The approximation using differentials was a bit larger than the actual change, but still fairly close.

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Would you like to explore further details, or have any questions? Here are some related questions:

1. How do differentials help estimate small changes in functions like profit?
2. What is the difference between the differential approximation and the actual change?
3. How can we improve the accuracy of the differential approximation?
4. What would happen if the change in $x$ were larger than one unit?
5. How do you find the derivative of a cubic function?

Tip: The smaller the change in $x$ (the closer $dx$ is to zero), the more accurate the differential approximation becomes.